Solve the following: 1. 3x - 7 = 14 2. x - 3(2-x) = 4x - (x - 1) 3. Identify the x-intercept and y-intercept of the line 4x - y = 12 4. Find the slope of the line joining the... Solve the following: 1. 3x - 7 = 14 2. x - 3(2-x) = 4x - (x - 1) 3. Identify the x-intercept and y-intercept of the line 4x - y = 12 4. Find the slope of the line joining the points (-1, 3) and (4, -3). 5. The system of linear equations \[\begin{cases}y = -\frac{3}{2}x + 10 \\ y = \frac{2}{3}x + 10\end{cases}\] has

Question image

Understand the Problem

This image contains 5 questions; all are math problems that require step-by-step solutions. Here's the breakdown:

  • Question 1: Solving a linear equation for x.
  • Question 2: Solving a linear equation for x with distribution.
  • Question 3: Finding the x and y intercepts of a linear equation.
  • Question 4: Determining the slope of a line given two points.
  • Question 5: Identifying the nature of solutions for a system of linear equations.

Answer

1. $x = 7$ 2. $x = 7$ 3. x-int: $(3,0)$ & y-int: $(0, -12)$ 4. $m = -\frac{6}{5}$ 5. a single point solution
Answer for screen readers
  1. (c) $x = 7$
  2. (b) $x = 7$
  3. (a) x-int: $(3,0)$ & y-int: $(0, -12)$
  4. (b) $m = -\frac{6}{5}$
  5. (b) a single point solution

Steps to Solve

  1. Solve for x in Question 1: $3x - 7 = 14$

Add 7 to both sides of the equation:

$3x - 7 + 7 = 14 + 7$

$3x = 21$

Divide both sides by 3:

$\frac{3x}{3} = \frac{21}{3}$

$x = 7$

  1. Solve for x in Question 2: $x - 3(2-x) = 4x - (x-1)$

Distribute the -3 on the left side and the -1 on the right side:

$x - 6 + 3x = 4x - x + 1$

Combine like terms on both sides:

$4x - 6 = 3x + 1$

Subtract $3x$ from both sides:

$4x - 3x - 6 = 3x - 3x + 1$

$x - 6 = 1$

Add 6 to both sides:

$x - 6 + 6 = 1 + 6$

$x = 7$

  1. Find the x and y intercepts in Question 3: $4x - y = 12$

To find the x-intercept, set $y = 0$ and solve for $x$:

$4x - 0 = 12$

$4x = 12$

$x = \frac{12}{4} = 3$

So, the x-intercept is $(3, 0)$.

To find the y-intercept, set $x = 0$ and solve for $y$:

$4(0) - y = 12$

$-y = 12$

$y = -12$

So, the y-intercept is $(0, -12)$.

  1. Find the slope of the line in Question 4, given points $(-1, 3)$ and $(4, -3)$

Use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

$m = \frac{-3 - 3}{4 - (-1)}$

$m = \frac{-6}{5}$

  1. Determine the nature of solutions for the system of linear equations in Question 5: $y = -\frac{3}{2}x + 10$ $y = \frac{2}{3}x + 10$

The slopes of the two lines are $-\frac{3}{2}$ and $\frac{2}{3}$. Since the slopes are different, the lines are not parallel and will intersect at one point. Since the y-intercepts are the same, the intersection will occur at the y-intercept. Therefore, there is a single point solution.

  1. (c) $x = 7$
  2. (b) $x = 7$
  3. (a) x-int: $(3,0)$ & y-int: $(0, -12)$
  4. (b) $m = -\frac{6}{5}$
  5. (b) a single point solution

More Information

The questions cover basic algebra concepts such as solving linear equations, finding intercepts, calculating slopes, and understanding systems of linear equations. These are fundamental skills in mathematics and are essential for more advanced topics.

Tips

  • Question 1: A common mistake is not performing the correct order of operations (addition before division) or making errors during arithmetic.

  • Question 2: Forgetting to distribute the negative sign correctly is a frequent error when simplifying the equation, as is sign errors more generally.

  • Question 3: Mixing up the x and y intercepts, or incorrectly setting x or y to zero is a common error.

  • Question 4: Swapping the order of subtraction in the numerator or denominator when calculating the slope can lead to an incorrect sign.

  • Question 5: Incorrectly interpreting the slopes and y-intercepts of the system of equations may lead to wrong conclusions about the nature of the solutions.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!
Use Quizgecko on...
Browser
Open
Browser
Browser